Canot Cycle, Work done

Hi. I have a qualitative question about the work done from state 1 to state 2 of a Carnot cycle. This is the Isothermal Expansion Phase. Is work done by the engine on the environment during THIS PHASE ONLY equal to the energy extracted from the hot reservoir to keep the temperature constant. I feel that it should be less. Is my reasoning incorrect?

Hi. I have a qualitative question about the work done from state 1 to state 2 of a Carnot cycle. This is the Isothermal Expansion Phase. Is work done by the engine on the environment during THIS PHASE ONLY equal to the energy extracted from the hot reservoir to keep the temperature constant. I feel that it should be less. Is my reasoning incorrect?

The work done is the area under the PV graph for the isothermal expansion phase. Since the isotherm is T = PV/nR = constant, it is of the form P = K/V where K is constant = nRT. The area under that graph ([itex]\int PdV = K\int dV/V[/itex]) is nRTln(Vi/Vf).